PCM Transmission in the
Exchange Plant
By M. R. AARON
(Manuscript received July 12, 1961)
Attention is focused on the choice of a code for transmitting a PCM signal
in the exchange plant via paper-insulated cable pairs. Inter- and intra-
system crosstalk via near-end coupling is the principal source of inter-
ference. A code and repeater structure for reconstructing this code to mini-
mize the effect of crosstalk into the timing channel of a reconstructive repeater
is emphasized. It is shown that the conventional self-timed unipolar repeater
is not suited to this environment. Several pseudo-ternary codes are developed
and surveyed for overcoming crosstalk interference. A bipolar code and a
repeater for reconstructing this code are chosen for a variety of reasons.
The repeater features nonlinear timing wave extraction and complete re-
timing and pulse width control, and its realization is discussed in detail in
a companion paper by J. S. Mayo. 3
I. INTRODUCTION
The choice of a modulation method or code for processing a signal
preparatory to transmission over a communication channel depends on a
multiplicity of conflicting parameters. All of the technical factors must
be considered to arrive at a system that serves a need at a justifiable cost.
Generally a few of the characteristics of the transmission medium stand
out to limit performance. In the situation at hand — pulse transmission
at high rates in the exchange plant — near-end crosstalk between cable
pairs is the principal transmission deterrent. Examination of the fac-
tors that enter into the choice of a simple code and a repeater to recon-
struct this code, to combat crosstalk interference, is the principal objec-
tive of this paper. Stated another way, we are seeking a transmission
scheme that maximizes the number of pairs in a cable that can be used
without pair selection.
99
100 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1962
1.1 Road Map
Before we attack this objective it is profitable for later comparisons to
provide some system background, to review the functions involved in a
PCM reconstructive repeater, and to define terminology. Though the
bulk of this introductory material has been covered in the literature, we
introduce some new approaches and concepts. It is essential to a clear
definition of the problem and our chosen solution. Most of this material
is covered in Sections 1.2 through 2.4. Some preliminary comparisons
are made in Section 2.5 that contribute to the choice of complete retim-
ing and regeneration. The crosstalk problem is brought into focus in Sec-
tion III. In Section IV it is shown that the conventional pulse-absence
of pulse method for transmitting binary PCM is not suited to the pres-
ent application. Section V contains a survey and evaluation of various
pseudo-ternary codes that minimize the problem of crosstalk into the
timing channel of a repeater. Reasons for choosing a self-timing bipolar
repeater are enumerated at the end of this section. Section VI is devoted
to a few words on equalization and equalization optimization. The final
section deals with some implications of the interaction of the bipolar
repeater with the actual exchange plant.
1.2 System Background
1.2.1 General
Other papers * - 2 in this issue have shown that the signal to be trans-
mitted over 22-gauge paper-insulated cable pairs is a 1.544-mc pulse
train. Repeaters to reconstruct this pulse train are spaced throughout the
medium with spacing dictated by economics, near-end crosstalk coupling
and impulse noise. Since loading coils must be removed from voice-fre-
quency pairs prior to pulse transmission, it is economically and adminis-
tratively desirable simply to replace the loading coils with repeaters. The
most common loading coil spacing is nominally 6000 feet, thereby dictat-
ing a repeater spacing equal to an integer multiple of 6000 feet. Twelve
thousand foot spacing is precluded by near-end crosstalk considerations,
as seen later. Therefore the nominal spacing between line repeaters is
6000 feet. Effective variation of electrical length about the nominal is
constrained to be ±500 feet by the use of line-build-out networks. The
importance of this assumed constraint on repeater spacing cannot be
overemphasized. In effect it removes repeater spacing as a system pa-
rameter to be used to combat crosstalk as far as this study is concerned.
For example, with shorter repeater spacing the crosstalk problem could
PCM TRANSMISSION IN THE EXCHANGE PLANT 101
be substantially reduced. The economic penalty could be countered by
raising the channel capacity above 24 channels. This of course, would
increase the bit rate and bring the crosstalk problem back into promi-
nence. The choice of a transmission scheme most tolerant to crosstalk
under this new situation might differ from that chosen for the GOOO-ft
repeater spacing.
In the vicinity of a central office another source of interference becomes
important, namely impulse noise due to switching transients coupled
from pair to pair via crosstalk coupling. To combat impulse noise, re-
peaters adjacent to an office are spaced nominally 3000 feet from the
office.
It should be emphasized that with the above spacings and practical
signal levels, thermal noise is not a problem.
1.2.2 Reconstructive Repeaters
It is well known that the salient feature of PCM transmission is the
ability to reconstruct the transmitted pulse train after it has traveled
through a dispersive, noisy medium. We will call such repeaters recon-
structive repeaters, as opposed to the more commonly used name of
regenerative repeaters, since regeneration in the circuit sense is only one
facet in reconstructing pulse trains. There are basically three functions
that must be performed by such a repeater, namely the three R's —
reshaping, retiming, and regeneration* This operational breakdown is
depicted schematically in Fig. 1 (a) where the pulse train is traced from
repeater to repeater. For purposes of illustration, we assume that the
transmitted pulse train at (b) consists of a series of pulses and spaces
representing the binary 1 and respectively. This is called a unipolar
pulse train, and eventually we will show that this type of code is not
suited to the exchange plant environment. However, at this point it
suffices for our simple explanations. After transmission over 6000 feet
of cable, the high-frequency content of each pulse is severely attenu-
ated and the received pulse train is corrupted by additive interference.
The spread-out, "noisy" train appears at (c) (cable output).
The primary function of the first repeater block, reshaping, is to shape
the signal and raise its level to the point where a pulse vs no pulse de-
cision can be made.
This process involves the inevitable compromise between interference
* Pulse regeneration as defined by t lie American Standards Association in ASA
C-42, Group 65 (definition 1)5.02-102) includes all of the three R's. However, for
our purposes the three-way split is more convenient.
102
THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1962
OUTPUT FROM
n TH REPEATER
BLOCK SCHEMATIC OF PULSE TRANSMISSION LINK IN PCM
IDEALIZED PULSE TRAIN AT (d)
WITHOUT INTERFERENCE
-WITH TIMING
WAVE
WITHOUT TIMING
WAVE
P/2 INPUT LEVEL
TRANSFER CHARACTERISTIC
OF IDEAL REGENERATOR
Fig. 1 — PCM transmission: (a) block schematic of pulse transmission link in
PCM, (l>) idealized pulse train at (d) without interference, (c) transfer character-
istic of ideal regenerator.
reduction and pulse resolution.* We will examine reshaping in minor
detail in this paper. A more extensive coverage is given in the paper by
Mayo, 3 which goes into the details of the realization of the bipolar re-
peater.
Final reconstruction of the pulse train at d (point of decision) is ac-
complished by the simultaneous operation of regeneration and retiming.
For purposes of this preliminary discussion we will assume that the re-
generator takes the form of a simple threshold detector. The regenerator
is enabled when the incoming signal plus interference exceeds the thresh-
old and when the timing wave at the output of the retimer has the proper
amplitude, or polarity. Other types of regenerators in which regeneration
is only partially accomplished in a single repeater are known as partial
regenerators. 4 Several partial regenerators in tandem approach complete
regeneration. Partial regeneration will not be considered further since
complete regeneration can be approached closely with available circuitry
(i.e., blocking oscillators) at the frequencies of interest. This situation
does not prevail at microwave frequencies. 4
The purpose of the retimer is threefold: (1) to provide a signal to
* Readers schooled in the lore of digital transmission will recognize the re-
shaping network as that circuit which, in conjunction with the line and trans-
mitted pulse shape, determines the "eye" picture, i.e., a snapshot of all possible
pulse combinations that contribute to each time slot. See Ref. 3.
PCM TRANSMISSION IN THE EXCHANGE PLANT 103
sample the pulse train where its peak is expected, i.e., where the signal-
to-interference ratio is host, (2) to maintain the proper pulse spacing
and reduce pulse jitter to minimize distortion in the receiving terminal,
and (3) to allow the timing signal to he used to turn off the regenerator
to maintain proper pulse width.
In the above manner, provided the signal-to-interference ratio is
sufficiently high and the time jitter sufficiently low, the reconstructed
pulse train at e is almost a replica of the original.
II. TIMING
Since timing plays an important role in PCM transmission, we will
introduce and classify methods for launching, extracting and using the
timing wave. In addition, we will summarize the sources of timing jitter.
Finally, we will make a preliminary comparison of various timing tech-
niques with respect to sources of jitter exclusive of crosstalk. After the
initial comparison is made we will characterize the medium and concen-
trate on the crosstalk problem.
2.1 Timing Generation
Methods for launching and extracting timing information can be
classified broadly according to whether or not an extra pair is used for
transmitting the timing signal. These broad divisions are further sub-
divided in Table I.
The self-timing category in Table I has been explored in several
papers. For this class, the timing wave is obtained by processing the in-
formation-bearing pulse train by either linear or nonlinear means or
both. Linear extraction, as exemplified by the work of Wrathall, 5 Sunde/'
and Bennett, 7 is used when the power spectral density of the transmitted
pulse train has a discrete line spectrum with a component at the pulse
repetition frequency. We will display other pulse trains in which a dis-
crete line at submultiples of the bit rate is produced that can be used
Table I — Timing Classification
1. Same-pair timing
a. Timing from information-bearing pulses, generally called self timing.
(1) Linear extraction — forward or backward
(2) Nonlinear extraction — forward or backward
b. Timing wave added
(1) Without spectral null
(2) With spectral null
c. Hybrid or dual -mode timing
2. Separate-pair timing
104 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1902
for timing. In other words, the pulse train may have a periodic average
value with a fundamental component at frequencies below the bit fre-
quency. This timing component is obtained by exciting any one of several
circuits or combinations thereof with the transmitted pulse train. Cir-
cuit approaches will be mentioned later.
Self timing can also be employed when the discrete line at the bit
rate is absent. Bennett 7 discussed one method for achieving this, and we
will consider other possibilities in a later section. Obviously, nonlinear
techniques must be used here that take advantage of the underlying
phase structure of the pulse train.
When the timing wave is obtained from the transmitted pulse train,
it is known as forward-acting. Backward-acting timing indicates that
the timing wave is obtained from the reconstructed pulse train at the
repeater output and fed back to an internal point in the repeater. We will
cover the relative merits of these two approaches when we discuss meth-
ods for using the timing wave.
Category (b) under same-pah- timing in Table I has not been covered
previously in the literature. In this approach a separate sinusoidal sig-
nal is added to the transmitted or reconstructed pulse train and extracted
at the next repeater. The added timing wave may be used to augment
the timing component inherent in the pulse train to insure adequate
timing when the pulse train is sparse. Alternatively, the added timing
wave can assume the entire timing burden. This may be achieved by
any of several methods. First, the energy in the pulse train in the neigh-
borhood of the bit rate can be eliminated by an appropriate filter prior
to the addition of the timing wave. Another approach is to use a pulse
transmission scheme in which the power spectral density has a null at
the bit frequency. A timing wave can be inserted in the resulting slot.
Spectral nulls at the bit rate may be obtained by pulse shaping or by
converting the binary pulse train to a three-level code, as we shall dem-
onstrate. With the latter method it is possible to produce spectral nulls
at submultiples of the bit frequency.
The last same-pan timing category of Table I is really a transition
class that leads into separate timing. Dual-mode timing involves the use
of self timing in one direction of transmission and the simultaneous use
of this timing wave for timing the other direction of transmission. The
slave system can use a transmission scheme that has a spectral null at
the timing frequency. This approach eliminates the near-end crosstalk
(NEXT) interference at the timing frequency.
The final category in Table I is self-explanatory. Further comparison
of all these schemes is deferred until we can bring some other factors to
bear.
PCM TRANSMISSION IN THE EXCHANGE PLANT
105
2.2 Methods of Using Timing Wave
Just as in the case of regeneration, retiming can be classified as com-
plete or partial. In this section these and related terms, as well as some
of those used previously, are given more concrete significance with the
aid of the diagrams in Fig. 2.
The simplest scheme to instrument is the partial retiming approach
used by Wrathall and analyzed by Sunde. In this approach, the peak
of the recovered timing wave (clock) is pinned to ground or some other
convenient reference and added to the incoming pulse train, as shown in
Fig. 2(a). When the signal-plus-timing wave exceeds the threshold level
the regenerator is fired. Obviously, as the timing wave amplitude be-
comes larger, sampling of the input pulse occurs closer and closer to the
pulse peak where the signal-to-interference ratio is best. The timing wave
OUTPUT-INPUT CHARACTERISTIC FOR
REGENERATION (IDEAL)
'/2 R + P
INPUT
a) PARTIAL RETIMING
R >0
R<
1/2
INPUT
(b) PERMISSIVE TIMING (WITH PULSE WIDTH CONTROL)
• V.W.Wr
OUTPUT
R >0
R <
1/2
INPUT
(C) COMPLETE RETIMING (WITH PULSE WIDTH CONTROL)
Kip;. 2— Retiming methods: (a) partial retiming, (b) permissive timing (with
pulse width control), (c) complete retiming (with pulse width control).
106 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1932
may be obtained from either the incoming or regenerated pulse train
or from a separate pair.
Another method for using the timing wave is illustrated in Fig. 2(b).
We call this permissive timing with pulse width control. Both the in-
coming pulse train and the clock are put into an "and" gate. When the
signal pulse exceeds the threshold and the clock is in its positive half
cycle, standard current is fed to a regenerator (a blocking oscillator, for
example) and the regeneration process is started. When the clock wave
enters its negative half cycle, current is extracted from the regenerator
to turn it off. In this manner the pulse width is constrained to be less
than one-half cycle of the clock.
One final approach is depicted in Fig. 2(c). This approach is known as
complete retiming with pulse width control. In this method, narrow
pulses are generated at the positive-going or negative-going zero cross-
ings of the timing wave and used for gating the incoming pulse train.
Similarly, pulses generated at the negative-going or positive-going zero
crossings are used to turn off the regenerator to control the width of the
regenerated pulses. In passing we note that the narrower the sampling
pulse, the smaller the fraction of time that interference is effective in
obscuring a pulse decision. The importance of the width of the sampling
pulse in combating crosstalk is discussed in the paper by Mayo. 3
As discussed by Sunde 6 and Rowe, 10 complete retiming precludes
timing from the reconstructed pulse train. Backward-acting timing is
also taboo with permissive timing.
2.3 Timing Wave Extractors
In self-timed repeaters, several circuit configurations have been in-
vestigated for timing-wave extraction. Since the timing recovery prob-
lem is similar to that involved in synch recovery in TV, it is to be ex-
pected that circuit approaches investigated for that application should
be pertinent here. 8 They include:
1. Tuned circuits
a. single tuned — LC
b. double tuned — LC and other narrow-band filters
c. crystal filters
2. Controlled passive filter
3. Clutched (locked) oscillator
4. Phase-locked oscillator — APC loop
5. Phase- and frequency-locked oscillator — dual mode.
By far the simplest and cheapest circuit is the single tuned LC tank.
PCM TRANSMISSION IN THE EXCHANGE PLANT 107
Most of the other possible implementations have been examined for
this application.
Detailed discussion of the merits of each approach is beyond the scope
of this paper. It suffices to point out that an automatic phase-control
(APC) loop is the ideal timing wave extractor, and an APC loop for
this application has been constructed. This relatively complicated cir-
cuit is available, if required, to mop-up timing deviations that propagate
through a chain of repeaters with simple LC tanks. For the short re-
peater chains encountered in the exchange cable application, the LC
tank with a loaded Q of 100 can be designed to meet jitter requirements,
system economics, and space requirements. Therefore, in the remainder
of the paper we will concentrate on this simple implementation.
2.4 Sources of Timing Jitter
In self-timed reconstructive repeaters using LC tanks for timing re-
covery, several sources of timing jitter and mistiming arise. They are
(1) thermal and impulse noise, (2) mistiming, (3) finite pulse width,
(4) amplitude-to-phase conversion in nonlinear devices, and (5) cross-
talk.
2.4.1 Thermal and Impulse Noise
De Lange and Pustelnyk 9 have shown that thermal noise is not a
serious contributor to timing jitter in the timing channel of a recon-
structive repeater. Stated another way, if thermal noise is small enough
such that the pulses can be recognized with low probability of error with
no noise in the timing channel, then the addition of noise in the timing
channel results in a negligible increase in the error probability. Physically
this is to be expected since the narrow-band timing extractor accepts
only a small fraction of the noise power for circuit Q's of the order of
100. This conclusion is equally valid for impulse noise for the same rea-
son as above and also because the average rate of occurrence of impulses
is considerably less than the bit rate or the minimum pulse density al-
lowable in the system.
At this juncture it is appropriate to point out that the timing wave
amplitude must exceed a certain minimum value to operate the timing
gate and to limit some of the sources of mistiming to be discussed below.
This presents a limitation on the minimum pulse density that can suc-
cessfully be reconstructed in a repeater string for the transmission
schemes of type la in Table I. It should be emphasized that this limita-
108 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1902
tion will prevail with any of the circuit implementations in a practical
environment that includes timing recovery with a mistuned extractor
excited by a baseband pulse train that permits spaces. Furthermore,
this lower bound is a function of the statistics of the signal being trans-
mitted, the allowable error rate, the kind of interference, the effective
Q of the timing circuit, and the amplification following the timing tank.
Once the timing wave has reached a sufficiently large amplitude, the
higher the Q, the longer the gap between pulses that can be bridged.
However, the higher the Q, the greater the phase slope of the tuned
circuit and the smaller the allowable mistuning for a given signal-to-
interference ratio. This inherent compromise between Q and mistuning
will be discussed quantitatively below. In the exchange carrier applica-
tion, it is necessary to limit the maximum number of spaces between
pulses to about 15. This is achieved by eliminating the all-zeros code in
the encoder. Attendant to this constraint is a small change in terminal
distortion due to clipping.
2.4.2 Mistuning
When the tuned circuit is mistuned from the p.r.f. (pulse repetition
frequency), the information-bearing pulses are sampled away from their
peaks. This lowers the interference allowable for a specified error rate.
As noted above, the higher the Q, the larger the shift in the zero-cross-
ings of the timing wave at the output of the tuned circuit for a fixed
mistuning. However, a high Q is desirable to bridge gaps and reduce
timing jitter due to finite pulse width, as discussed below. Furthermore,
the effect of mistuning is dependent upon whether forward- or back-
ward-acting timing extraction is used, as well as whether partial or
complete retiming is employed. We will shortly compare all of these
techniques quantitatively.
2.4.3 Finite Pulse Width and Pattern Effects
As shown by Rowe, 10 when the pulses exciting the tuned circuit are
not impulses or 50 per cent duty cycle rectangular pulses, the zero-
crossings at the output of the tuned circuit are perturbed from their
nominal positions. These deviations are dependent upon the pulse den-
sity (or pattern), the Q of the tuned circuit, and the shape of the pulses
exciting the tank; and they differ for positive- and negative-going zero-
crossings. This is a form of amplitude-to-phase conversion analogous to
the amplitude-to-phase conversion in FM limiters.
Pattern jitter also occurs in partial retiming due to the above effects.
PCM TRANSMISSION IN THE EXCHANGE PLANT 109
While backward-acting timing with ideal 50 per cent duty cycle rec-
tangular pulses exciting the tuned circuit eliminates the finite pulse
width effect, it does suffer from severe pattern jitter due to variation
in timing wave amplitude with pulse density.
2.4.4 Nonlinear Amplilude-lo- Phase Conversion
Any of the nonlinear circuits either preceding or following the tuned
circuit will inevitably perturb the zero-crossings of the timing wave. In
addition the regenerator, nominally a blocking oscillator, will not be an
ideal zero-memory nonlinear device. Consequently, the spacing between
reconstructed pulses will be altered by its inherent storage and will be
dependent on the past history of the reconstructed pulse train.
2.4.5 Crosstalk
Intra- and inter-system crosstalk into the timing channel of a for-
ward-acting repeater, either partially or completely retimed, results in
a shift of the zero-crossings of the timing wave. We will devote the bulk
of the paper to this problem.
2.5 Preliminary Comparisons
At this point, we will make preliminary quantitative comparisons of
some of the methods of extracting and using the timing wave in a self-
timed repeater. Our attention will be confined to the effects of mistuning
and pattern jitter.
2.5.1 Mistuning
As noted previously, when the tuned circuit is mistimed from the
pulse repetition frequency, tolerance to interference is reduced. To com-
pare the various self-timing schemes quantitatively, we will make the
following assumptions:
1. The information-bearing pulses are raised cosine pulses of base
width, either T, 1.57', or 2T* wide.
2. The output of the tuned circuit is a sinusoid whose peak-to-peak
amplitude is equal to the height of the information-bearing pulse. For
partial retiming, the positive peak is clamped to ground and added to
the signal.
* In the 27' cusp, the discrete component at the bit rate disappears. Therefore,
nonlinear methods must be used to create a discrete line at the bit frequency for
purposes of timing.
110 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1962
Under these conditions, we can use the methods developed by Sunde 6
to compare backward and forward-acting partial retiming with com-
plete retiming. In the latter case, we assume that the phase shift in the
zero crossings of the timing wave is given by its average value, approxi-
mately 2Q(A//0 ; A/// is the fractional mistiming of the tuned circuit.
Fig. 3 shows the reduction in allowable interference in the presence of
mistuning for the three self-timing methods of interest. As expected,
forward-acting partial retiming and complete retiming arc far superior
to backward-acting partial retiming. This is particularly noticeable for
the case where the pulse width is from 1.5 to 2 time slots wide
at its base. It will be shown that a pulse shape in this region is
required to minimize the effects of crosstalk interference in making the
pulse-no pulse decision. Based on the fact that there are several sources
of impairment that must share the allowable margin against error, that
portion of the margin allocated to mistuning must be as small as pos-
sible, consistent with presently available components. In the present
state of the art, including initial misplacement and aging, it appears
that the maximum phase shift in the tuned circuit (LC tank) can be
held to about ±30° for a Q of about 100. This corresponds to 2Q(Af//)
= 0.6, or Af/f = 0.003. From Fig. 3 and the fact that equalization to
minimize crosstalk interference must yield a pulse close to the 1.5 to
2T cases, it can be seen that backward-acting timing should not be used
for this application.
It should be pointed out that there is nothing restrictive about the
use of raised cosine pulses in making this point. It can be shown that
the conclusion arrived at above remains valid for other similar pulse
shapes, time limited or not, and for other practical ratios of timing wave
amplitude to pulse peak. Indeed, for larger ratios of timing wave ampli-
tude to pulse peak, the stability problem is further aggravated. This is
due to the positive feedback nature of the timing loop in backward-
acting timing. A straightforward extension of Sundc's work will verify
this contention. In addition, experimental work by A. C. Norwine*
confirms this expected behavior.
2.5.2 Finite Pulse Width and Pattern Effects
In this category, we will compare partial retiming (forward-acting)
with complete retiming for both periodic and random pulse patterns.
With periodic patterns, both raised cosine and Gaussian pulses will be
considered. Several ratios of average timing wave amplitude to peak
* Private communication.
PCM TRANSMISSION IN THE EXCHANGE PLANT
111
^ FOR Q = iOO
Fig. 3 — Effects of mistiming.
pulse height will be considered with partial retiming. We continue to
consider binary (1,0) pulse trains.
2.5.2.1 Periodic Pulse Patterns — Partial Retiming. To divorce pat-
tern effects from mistiming, we assume that the peak of the timing wave
occurs at the pulse peak. Further, the positive peak of the timing wave
is pinned to ground and the timing wave amplitude is given by
2tt/ n
an
2J7
1 — cos
T
(1)
where
a is the peak-to-peak amplitude of the wave when all pulses are pres-
ent; i.e., n = 1 = M, and
n is the number of pulses that occur in an ilf-bit word.
When n t^ 1, we assume that the pulses arc adjacent, and that the
time slot examined is the one containing the last pulse in the group. In
the following, we will specialize to n = 1, and M will vary from 1 to 8.
112 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 19G2
The assumptions underlying (1) are, first, that the pulses exciting the
tuned circuit are very narrow pulses obtained by processing the incom-
ing signal, and second, that the timing wave amplitude in a real repeater
is dependent upon the density of pulses exciting the tuned circuit. While
the first assumption does not correspond to the practical case, it does
give an optimistic; view for partial retiming. We will find that even with
this idealization, partial retiming is considerably inferior to complete
retiming in maintaining the proper pulse spacing under both steady-
state and random conditions. It is possible to drop the first assumption
here, as we do for complete retiming. The additional analysis simply
lends further emphasis to the conclusion.
Two pulse shapes will be considered. First, the raised cosine with
P( = l(l + co S ^) where | , | < £ ^
= elsewhere.
The pulse width is T/s. We consider s = 1, f. Secondly,
p(t) = e -(.*/In 2 )(/ 6 0*_ (3)
In (3), / 6 is the frequency at which the transform of the Gaussian pulse
is 6 db down from its low-frequency asymptote. Alternatively, we can
write the Gaussian pulse as
P(t) = e -*w^\ (4)
In (4), T w is the pulse width between points where the pulse amplitude
is 0.1 of its peak.
In partial retiming, the incoming pulse is regenerated at the instant
when the sum of the signal plus the timing wave exceeds the slicing level,
which is assumed to be at \. The resulting problem can be solved graph-
ically, iteratively, or, for some cases, analytically. Fig. 4 displays the
results of these computations for raised cosine pulses.* The ordinate
gives the steady-state phase shift in degrees, measured from the pulse
peak, as a function of the pulse pattern. In all cases we have subtracted
out the phase shift corresponding to all pulses present. The important
point is the fact that the difference in phase shift between all pulses
present and | present is quite large: about 60° for the widest pulse and
50° for a pulse width of 1.55P. Fig. 5 presents the same information for
Gaussian pulses, and the results are not substantially different from
those obtained for raised cosine pulses. It is worth noting that inter-
* Similar computations have been made in an unpublished memorandum by
W. M. Goodall and O. E. De Lange.
PCM TRANSMISSION' IN THE EXCHANGE PLANT
113
Z -20
Yosts'
/
/ /
>^i
i
f
RAISED COSINE PULSE
BASE WIDTH = 2T
COMPLETE RETIMING
(POSITIVE AND
NEGATIVE- GOING
ZERO CROSSINGS
ABOUT THE SAME)
a = 2 (VERY LITTLE
DIFFERENCE FOR
OTHER VALUES OF a)
RAISED COSINE PULSE
BASE WIDTH = I.5T
1 1 1
1 8 4
1
2
PULSE PATTERN
Fig. 4 — Pattern effects
width = 1.5T.
raised cosine pulse: (a) base width = IT, (b) base
ference riding on the leading edge of the pulse will result in additional
timing jitter.
2.5.2.2 Periodic Pulse Patterns — Complete Retiming. In complete
retiming, timing jitter results from the finite width of the pulses exciting
the tuned circuit and the variation of pulse density. Using a minor mod-
f
-——"*■"■■" ^ < ^<^ > ^
20
/
/
*~*/ //
/
/
f
10
1
I
I
COMPLETE
RETIMING
PARTIAL
RETIMING
60
/I
II
f
GAUSSIAN PULSE
f 6 = 0.75MC
ORT w =l.7T
flO
/
,'p
/
\ 8 = 2
(LITTLE DIFFERENCE
FOR OTHER
VALUES OF a)
N = NEGATIVE -GOING
ZERO CROSSINGS
P = POSITIVE -GOING
ZERO CROSSINGS
GAUSSIAN PULSE
f 6 = 0.9MC
OR T w = 1 .4 T
I 8 4
PULSE PATTERN
Fig. 5 — Pattern effects — Gaussian pulse: (a) / 6 = 0.75 mc, (b)/ 6 = 0.90 mc.
114 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1962
ification of Rowe's method, we have displayed the phase shift in the
zero crossings at the output of the tuned circuit in Figs. 4 and 5. A Q
of 100 is assumed, and both positive- and negative-going zero-crossings
are considered. It can be seen that the phase shift in this case is con-
siderably smaller than in the case of partial retiming. Since these figures
display static performance, we cannot immediately conclude that the
resulting distortion in the reconstructed analog signal will be either
tolerable or intolerable. Distortion at the output of the demultiplex
filter in the terminal due to phase modulation on the PAM samples is
dependent upon both the amplitude of the jitter and its rate of change.
These quantities in turn depend upon the bandwidth (Q) of the timing
extractor used in each repeater. Without going into detail, based on the
dynamic effects of timing jitter propagation due to pattern shift in a
string of repeaters, it can be concluded that partial retiming will be un-
satisfactory for this application. Furthermore, the additional circuitry
required in complete retiming over that needed for partial retiming
(forward) is relatively simple and cheap. Finally, and most important,
complete retiming makes pulse width control relatively easy, and the
use of narrow sampling pulses yields a substantial crosstalk advantage. 3
From Fig. 3, we see that wide pulses are desirable to minimize the
effects of mistiming, while Fig. 5 shows that narrower pulses are desirable
for minimizing jitter in the zero-crossings at the output of the timing
wave extractor. Furthermore the positive-going zero-crossings are per-
turbed to a larger extent than negative-going zero-crossings for sparse
patterns. This suggests separate equalization in the timing path to nar-
row the pulses prior to excitation of the tuned circuit. Slicing in the
timing path is useful in this regard and also serves to eliminate low-level
interference hi the absence of a pulse. This will be emphasized later in
connection with crosstalk.
2.5.2.3 Random Pulse Patterns. Another view of the dynamic effects
of pattern jitter may be obtained by determining the probability dis-
tribution of phase jitter for random pulse patterns. This approach will
be published in another paper 17 where we will include both mistuning
and finite pulse width effects. This information can be used to further
cement our choice of complete retiming for this application.
2.6 Interim Summary
At this point a short summary of our preliminary conclusions is in
order. Based primarily on mistuning and pattern effects, with a peek
ahead at crosstalk considerations, it is concluded that complete retim-
ing with pulse width control should be used for a self-timed repeater.
PCM TRANSMISSION IN THE EXCHANGE PLANT
115
This follows from the fact that our objective is to leave most of the mar-
gin against error to crosstalk interference in order to avoid pair selection.
For this same reason, and the fact that it is readily approached, com-
plete regeneration should be employed.
It should be apparent that pattern effects ideally can be eliminated
with timing wave added or by transmitting the timing wave on a sep-
arate pair. There are important economic and technical reasons why
these approaches are undesirable and we will cover them in Section V.
III. NATURE OF NEAR-END CROSSTALK
3.1 General
With the above preliminaries largely disposed of, we can concentrate
on the crosstalk problem. This problem arises when we consider both
directions of transmission for a single 24-channel system and is further
compounded when many 24-channel systems are transmitted on sep-
arate pairs in the same cable bundle. In effect, such a system is a com-
bination of time division and space division, and a typical repeater-to-
repeater link can be depicted in block diagram form as shown in Fig. 6.
fc 1 CABLE ! «
^-1--,"
Inext !
i r — i
I
— h-
, 1 ,
| !next|
! . l -t- j
i i
i • i ■
-+-> CABLE | *■
I L J
OTHER L.
NEXT AND
FEXT PATHS
-I FEXTI
I I
— *: 1 CABLE | T
' J I
._!_., >
! r 1
± f_i CABLE | *
| L 1
Fig. 6 — "Channel" for .V PCM systems.
116 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1962
We could write a matrix relationship between the transforms of the
voltages at the 2N output terminals and the 2N input terminals. The
elements of the matrix would contain the transfer function of the direct
cable paths on the main diagonal, and the transfer functions of the near-
end and far-end crosstalk paths would reside in the off-diagonal terms.
Such a representation, though elegant, would be of little use without a
detailed characterization of the elements of the matrix (channel) . At the
present state of our knowledge, the above viewpoint is not too useful.
The most important couplings fall in the category of near-end cross-
talk coupling (NEXT) between pairs operating in different directions of
transmission. A comprehensive treatment of the physical mechanism
from which crosstalk arises will not be attempted. A useful image of the
situation results from considering the 2N pairs (for N systems) as 2N
lossy coupled, tapped delay lines in which the "taps" (inductive and
capacitive coupling due to unbalances) have both random amplitude
and spacing. For our purposes, the macroscopic view obtained from
looking at the 4JV ports of this network will suffice. Furthermore, it will
be convenient to make a further abstraction from reality and define an
"equivalent crosstalker" and an "equivalent crosstalk path" to replace
the complicated statistical model given above. This approach will permit
us to attach a "crosstalk figure of merit" to each transmission scheme,
which will serve as a valuable aid hi sorting out those approaches most
tolerant to near-end crosstalk (NEXT).
3.2 NEXT Measurements
3.2.1 Single-Frequency Distributions
Over the past twenty years or so, hundreds of thousands of single-
frequency measurements have been made of crosstalk between pairs in
trunk cables. Relatively few of these measurements have been made in
the 100-kc to 10-mc region. A small number of measurements in this
frequency range have been made at Bell Laboratories by B. Smith on a
unit-constructed cable. We have used these preliminary data as a guide-
post to choose a transmission scheme.* The distribution of these meas-
urements at 1.5 mc (close to the bit rate of interest) is shown in Fig. 7.
Two cases of interest are shown. The worst case prevails when the inter-
fering pair is in the same unit as the pair into which it is crosstalking.
When the two pairs are in adjacent units, crosstalk coupling is reduced.
* It is to be emphasized that these are preliminary data and are included to
eonvey a feel for the magnitude of the problem; they do not necessarily typify
the average cable plant.
PCM TRANSMISSION IN THE EXCHANGE PLANT
117
The most favorable situation (not shown) occurs where opposite direc-
tions of transmission are located in diametrically opposed units of the
cable. Where unit integrity is maintained throughout the length of the
cable, it is obviously prudent to take advantage of the lowest crosstalk
coupling in installation. However, unit integrity is not universally ad-
hered to. Therefore, for purposes of discussion we will use the data on
Fig. 7 corresponding to the worst case.
It should be noted that, as shown, the distribution of pair-to-pair
crosstalk is distributed according to a log normal distribution. There is
some theoretical justification for this shape over most of the crosstalk
loss range. It is to be expected from physical considerations that the
distribution should truncate in the neighborhood of the tails. Indeed,
R. J. Herman in work at Bell Laboratories has shown that the log nor-
mal hypothesis is not supported by the data in this region.
3.2.2 Loss vs Frequency
Single-frequency crosstalk loss measurements are most useful for
comparing some transmission schemes with respect to crosstalk in the
o
<n!f>
wm
g<
Si
a. i
XH
ULu
aS
98
99.9
\
\
V
s
s
V WITHIN
N.UNIT
ADJACENT
UNITS
90 85 80 75 70 65 60 55 50 45
METALLIC-TO-METALLIC NEXT LOSS AT 1.5 MC
Kig. 7 — Preliminary near-end crosstalk loss distribution.
118 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1902
timing channel. However, these data are insufficient to define the effect
of crosstalk interference on the desired information-bearing signals. An
appraisal of the effect of crosstalk on the desired pulse train requires
the determination of the loss and phase of the crosstalk path or, alter-
natively, the pulse response of this path. Since the interfering signal is
made up of a combination of random pulse trains, and since the paths
they traverse to the system being interfered with also come from a ran-
dom family, a complete statistical description of the situation is ex-
tremely complicated and nonexistent. Measurements of the NEXT loss
between pairs, as a function of frequency, display a broad structure in
which the loss decreases with increasing frequency at about 4.5 db per
octave (coupling proportional to f 3 ' 4 ) for frequencies greater than about
200 kc. This is in good agreement with some unpublished theoretical
work of D. K. Gannett of Bell Laboratories. In addition to this nominal
smooth behavior as a function of frequency, measurements have revealed
other more rapid variations with frequency. This fine structure may be
attributed to relatively large localized capacitive and inductive un-
balances. These results are in accord with the model specified previously
as a visual aid and give a qualitative explanation of the classes of pulse
responses observed with pulse excitation of the crosstalk path.
3.2.3 Simplification
In our comparison of various transmission schemes, we will find it
convenient to neglect the fine structure associated with the crosstalk
path. We take this bold step with the obvious realization that we pre-
clude a completely definitive evaluation. It was mandatory to make this
abstraction from reality early in the system development when detailed
crosstalk data were not available and a development decision had to be
made. The principal advantage of this simplification is that it permits
us to isolate the classes of codes most tolerant to NEXT interference in
the timing channel of the disturbed repeater.
It was convenient and expedient to go one step further and replace
the multiplicity of crosstalk paths and interferers by a single capacitive
coupling and a single interferer for purposes of repeater design and ex-
perimentation. An equivalence between this gross simplification and the
real world can be determined empirically. Mapping of results obtained
in the simple domain onto the real world by analytical means is an ex-
tremely difficult chore. We will have a little more to say about this cor-
relation of models in a later section.
PCM TRANSMISSION IN THE EXCHANGE PLANT
119
x(f)
♦ RETIMER
Fig. 8 — "Equivalent" crosstalk situation.
IV. CROSSTALK INTO THE TIMING CHANNEL
4.1 Assumptions
With the simple "smooth" crosstalk model we can begin to be more
quantitative in our definition of the crosstalk problem, particularly with
respect to timing. For purposes of definition, and later comparisons, we
will consider the situation depicted in Fig. 8. The following assumptions
arc appropriate:
1. Nominal repeater spacing is 6000 feet on 22-gauge cable, and
"within-unit" distribution of metallic-to-mctallic NEXT is applicable.
The line loss corresponding to this length at 55°F is shown on Fig. 9.
0.04 0.06 0.1 0.2 0.4 0.6 0.8 1 2
FREQUENCY IN MEGACYCLES PER SECOND
Fig. !) — Lino loss and crosstalk loss.
120 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1962
In addition, the line loss under extreme temperature conditions (100°F),
under extreme manufacturing deviations (3tr limit), and line length of
6500 feet is also depicted. The mean crosstalk and the 1 per cent cross-
talk loss are also shown using both 6 db and 4.5 db per octave extrapo-
lations from the mean and 1 per cent values of Fig. 7.
2. A disparity of pulse densities of 7/1 on disturbing and disturbed
pairs is quite likely.
This assumption is of course intimately tied up with the code pattern
transmitted by the terminals and can arise when there are several idle
channels in the disturbing system and the coder is misbiased. Under
conditions of misbiasing, the coder may put out level 63 for the idle
circuit or no speech condition instead of 64. Clearly this also depends
upon noise riding on the reference. There are several techniques for
minimizing the effects of this disparity in pulse densities on repeater
timing, and they will be considered anon. An 8/1 disparity in pulse
densities can occur for speech transmission but with a low probability.
For transmission of high speed data over T-l lines, the disparity in
pulse densities can be 8/1 with significant probability.
3. Finally, we will determine the allowable crosstalk at the pulse
repetition frequency such that the desired timing frequency component
at the input to the tuned circuit is 6 db stronger than the timing fre-
quency component that traverses the crosstalk path. This corresponds
to a maximum of 30° phase shift in the timing signal relative to the
position of the desired pulse peak. It turns out that this amount of jitter
is tolerable as far as pattern jitter effects on the reconstructed PAM
signal are concerned for the bandwidths (Q = 100) of the tuned circuits
used in the repeaters. 3 We will not be concerned with these dynamic
effects in any detail. Our picture will, by the large, be a static one, and
the bandwidth of the tuned circuit will not enter. The allowable cross-
talk loss as determined above is defined as the "crosstalk figure of merit"
for a transmission scheme. The smaller this crosstalk figure, the larger
the number of pairs that can be used for PCM transmission.
From the data on Fig. 9, we can anticipate the difficulty associated
with transmitting timing information down the repeatered line at the
bit frequency. For example, if we assume the worst case line loss and
equal transmitted timing components on both disturbing and disturbed
pairs, then 53 + 6 = 59 db is the required crosstalk loss to meet the
30° phase shift requirement. With this crosstalk figure of merit, it is
apparent from the distribution of Fig. 7 that pair selection cannot be
ruled out even with a single interferer and no adverse disparity in pulse
densities.
PCM TRANSMISSION IN THE EXCHANGE PLANT 121
It should 1)0 quite evident from an extrapolation of Fig. 9 that 12,000-
I'ikiI spacing, twice the normal load-coil spacing, is an unworkable situa-
tion.
4.2 Pair Selection
Now that we have raised the specter of pair selection, it will be
instructive to bring the magnitude and scope of the associated measure-
ment problem to the fore. Fig. 6 gives a picture of the crosstalk situa-
tion at the extremities of a repeater-to-repeater link. For N PCM sys-
tems in the cable, there are N' 2 crosstalk exposures at each end of the
link. In an /.-mile route containing N PCM systems, approximately
2N 2 L exposures exist. As an example, consider the installation of 20
PCM systems, each 25 miles long. Approximately 20,000 exposures
exist, and the measurement, characterization and exclusion of pairs is
an enormous and expensive task. To avoid this problem we must choose
a transmission method with a high degree of tolerance to intra- and
inter-system crosstalk. More precisely, a transmission scheme sufficiently
insensitive to near-end crosstalk interference must be chosen such that
the probability of failure of a single repeater (due to this cause), out of
the approximately 2NL installed, is small. This calls for a low crosstalk
figure of merit.
4.3 Unipolar Pulse Train
The simplest and most common form for transmitting binary PCM
is to represent a one by a pulse and a zero by the absence of a pulse.
Under these conditions, with pulses and spaces uncorrected, it is well
known that the power spectral density of the pulse train is
, V/ ) = WH'pO-p) + | G(f) p £ £ , (/ _ , ]/r , (5)
1 1 n=— oo
In (5), p and (1 — p) are the probabilities of pulse and no-pulse respec-
tively, G(f) is the Fourier transform of the pulse shape, and T is the
reciprocal of the pulse repetition frequency /, . The presence of a discrete
line in the spectrum at the bit rate (assuming | G(f r ) | ^ 0) permits the
extraction of timing information from the pulse train by means of a sim-
ple tuned circuit. With a random pulse train on both disturbing and dis-
turbed pairs, the crosstalk figure of merit for worst case line loss is 59 db
as before. Even with nominal line loss the figure of merit is 49 db. In
either case, pair selection cannot be excluded for only one system. Un-
favorable disparity in pulse densities of 7/1 raises the worst case figure
122 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1962
of merit by 17 db to 76 db. From Fig. 7 we note that only 40 per cent of
the exposures have crosstalk loss exceeding 76 db. This shows clearly
that this simple transmission scheme is not suited to the exchange plant
environment. Obviously, the direction to proceed is to consider trans-
mission schemes in which the timing information is transmitted at a fre-
quency below the bit rate where the line loss is reduced and the crosstalk
loss is increased. In the succeeding sections we will present some methods
for achieving this end.
V. OTHER PULSE TRANSMISSION SCHEMES
5.1 General
All of the transmission schemes that we will consider have a common
thread. They involve a conversion of the binary train to a three-level
code (pseudo-ternary) for transmission over the line. Conversions are
accomplished on a bit-by-bit basis. Translation from the binary code to
a three-level code by operating on multiple bits or binary words will not
be discussed. In addition, hybrid combinations (series or parallel or
both) of the bit-by-bit converters are possible. However, the latter two
approaches have been ruled out for this application, cither on the basis
of economics or the fact that they do not afford substantial technical ad-
vantages over the simpler methods. It should be understood, however,
that the provision of a three-level reconstructive repeater permits the
utlization of some of the more complex code translators at the terminal
should unforeseen conditions dictate their choice.
Several of the pseudo-ternary pulse trains will be described prior to
placing them in the crosstalk environment. In addition, other factors will
be brought to bear in the comparison of the various three-level codes.
Consideration of these other factors involves judgments of the degree of
difficulty and the costs associated with the realization of the various ap-
proaches.
One of these additional factors involves low-frequency suppression.
Transformer coupling is required to couple an unbalanced repeater to
the balanced line and to provide a phantom path for remotely powering
the repeaters. Transmission of a pulse through transformers results in a
long transient undershoot that extends over several time slots and inter-
feres with subsequent pulses in the train. This of course reduces tolerance
to crosstalk. There are a host of techniques that have been used to com-
bat this low-frequency wander. A summary of most of these methods is
given in Ref. 11, and we do not intend to review them in this paper. We
PCM TRANSMISSION IN THE EXCHANGE PLANT 123
have found that many of the circuit approaches for minimizing or the-
oretically eliminating the effects of low-frequency suppression that ap-
pear ideal on paper have been found wanting when actually implemented
and placed in a real world environment. Pulse trains whose spectra con-
tain discrete lines at dc suffer most from low-frequency suppression.
This is another reason for discarding unipolar.
A second factor of interest involves compatability of PCM systems
with AM carrier systems using pairs in the same cable. Compatability is
a function of the number of PCM systems involved, the crosstalk loss in
the frequency region occupied by the AM system, and other details and
layout of the particular AM system. An indication of the relative com-
patability of the various PCM transmission approaches due to interfer-
ence from PCM into the AM system may be obtained from a comparison
of their power spectra. Since crosstalk loss decreases with increasing fre-
quency, the AM system that extends to the highest frequency will be
most affected. N carrier (its this picture. Therefore, we will use a fre-
quency close to the top of the N carrier band as a bench mark for com-
parison of the relevant power spectra. For convenience we choose
/ r /6 — 257 kc as the representative frequency. Since N carrier employs
frequency frogging, other higher frequencies are also of importance at a
high-low N repeater. In particular, frequencies up to 440 kc are of im-
portance. This is one reason why timing at/ r /4 does not seem attractive.
5.2 Time Polarity Control (TPC)
The first transmission scheme we consider is called time polarity
control. It was suggested and demonstrated by L. C. Thomas. In this ap-
proach, time slots are labeled alternately positive and negative. If a uni-
polar pulse occurs in a positively labeled time slot, it is transmitted un-
altered. On the other hand, if a unipolar pulse occurs in a time slot with
a negative label, the pulse is transmitted with negative polarity. Zeros
in the unipolar train are unaffected. The block diagram of a circuit for
achieving this end is shown in Fig. 10, along with an example of an ideal-
ized pulse train before and after conversion. Intuitively, discrete lines
in the power spectral density of the TPC train are expected at multiples
of half the bit rate. This inherent periodicity may be demonstrated by
computing the ensemble average of the TPC train. To do this, we assume
that the original binary pulse train has independent pulses and spaces
that occur with probability p and 1 — p. Furthermore, we assume that
the positive and negative pulse shapes in the TPC train are given by
0i (0 and —(MO, respectively. The latter assumption is introduced to
124
THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1962
UNIPOLAR
PULSES IN
II |0 |l |l
TPC
PULSES OUT
I |0 |l II
Fig. 10 — TPC converter.
account for practical differences in the two circuits that generate the
positive and negative pulses. Under these conditions, the ensemble av-
erage of the pulse train is
av x(t) = | E (fctt ~ 2nT) - g*(t - (2» + 1)T)). (6)
2i n=— «
That (6) is periodic with period 2T can be seen from
av x(t + 2T) = | E Mt - (2/i - 2)T) - g*(t - (2n - 1)T))
L n=—°
(7)
= av x(t)
by making a change in the summation index from n — 1 torn.
The power spectral density for TPC may be computed by any one of
several approachesf to give
P(f) c = ^^[\Cnn\G 2 \ 2 )
(8)
for the continuous portion of the spectrum. The discrete spectrum is
Pifh = ^ [I Gt I 2 + I G 2 1 2 - 2ReG 1 G 2 *e-*" r \ j£ 8 (f - ^) (9)
In the ideal case when Cn = G 2
r*(f) -
p(l - p) i a
ftf + Sslftl 1 2 «(/-
'2n -
_0A
(10)
From (10) it can be seen that the continuous part of the spectrum is
t Signal flow graph techniques 1213 were used in deriving the expressions for the
power spectral densities.
PCM TRANSMISSION IN THE EXCHANGE PLANT 125
identical with that of unipolar assuming the same pulse shape in both
cases. In ideal TPC discrete lines occur at odd integer multiples of one-
half the bit rate. This permits timing at half the bit rate. Reconstruction
of the original binary train at the receiver is achieved simply by rectifica-
tion of the TPC train consisting of non-overlapping pulses. Half bit rate
timing in the presence of a random crosstalk interferer with a crosstalk
loss of 39 db at half the bit rate yields a crosstalk figure of merit of
39 — 4.5 = 34.5 db for worst case line loss.
It is advisable for other comparisons to catalogue certain features of
the power spectrum of TPC. Unlike unipolar, TPC with balanced pulses
has no discrete line at dc. This eases but does not eliminate the low-fre-
quency suppression problem. It still is necessary to consider long runs of
alternating positive (or negative) pulses and spaces. Lack of dc trans-
mission will increase the susceptibility of the system to errors due to
crosstalk. DC restoration operating on both the positive and negative
peaks can be employed to reduce this effect. This is a relatively difficult
circuit problem and results in an increased repeater cost over other
methods to be considered.
There are several modifications of the basic TPC converter that can
be made to achieve other codes. Increasing the number of stages in the
counter produces higher-order TPC. For an il/-stage counter; M time
slots are labeled positive, the next M negative, and the cycle is repeated.
If the positive and negative output pulses are identical (except of course
for sign), the continuous spectrum of TPC-il/, is identical with unipolar.
Discrete lines appear at odd integer multiples oif r /2M. For M > 1, the
fundamental occurs at frequencies occupied by AM carrier systems and
precludes comparability between PCM and the AM system. Further-
more, more instrumentation is required hi the timing path of a repeater
to process the signal preparatory to performing complete retiming with
pulse width control.
5.3 Bipolar
Another and more useful modification of the converter of Fig. 10 re-
sults when the clock input to the counter stage is replaced by the incom-
ing unipolar train, as in Fig. 11. The output pseudo-ternary code is
thereby constrained such that two successive pulses, whenever they oc-
cur, must be of opposite sign. We assume the same conditions on the uni-
polar train used previously. Further, it is assumed that the transforms
of the positive and negative pulses are G and [— (1 + a)0 + CrJ respec-
tively. This brings out the differences in the two shapes more explicitly.
120 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1962
UNIPOLAR
PULSES IN
BIPOLAR
PULSES OUT
Fig. 11 — Bipolar converter.
In the ideal case (balanced pulses) a = = Gi . With the general as-
sumptions, the resulting expression for the continuous part of the power
spectrum is long and will be covered elsewhere. The discrete spectrum is
given in general by:
W)« = hi I ~ aG + Gl I' 2 *(/ - tfr).
4T
(11)
This obviously disappears under ideal conditions. Subject to this ideali-
zation, the spectrum contains only the continuous component given by:
Pz(f) =
p(l - v)
T
a
1 - cos uT
_1 + 2(2p - 1) cos wT + (2p -
D 2 J
(12)
For the same pulse shapes in unipolar and bipolar
W)-4tT^WW^] ?i(/) '' (13)
The subscript on 1\ indicates the continuous component of (5). With
50 per cent duty cycle rectangular pulses, the spectrum is shown in Fig.
12 (labeled N = I) for p = 1/2. From this figure or (13) it is apparent
that the spectrum has nulls at integer multiples of the bit rate (as well
as nulls where \G\ = 0). Absence of power at dc and the reduction of
the spectrum in the neighborhood thereof eases requirements on the
low-frequency performance of the transformers. This also follows from
the fact that no two successive pulses, independent of their spacing, can
be of the same sign.
In making this conversion we have suppressed all discrete components
that might be useful for timing. As shown in the Appendix, rectification
of the bipolar train produces discrete lines in the spectrum, and the com-
PCM TRANSMISSION IN THE EXCHANGE PLANT 127
ponent at the bit rate can be used for purposes of timing.* The question
arises as to where the energy for timing comes from. The analysis given
in the Appendix (Section A.l) shows that the timing component arises
largely from the region where the bipolar power spectrum is most con-
centrated. This is in the neighborhood of half the bit rate, and we will
use this frequency to characterize the "timing frequency" for this method
of transmission. f This result is intuitively satisfying since the rectifier
acts as a frequency doubler. A square law rectifier has been assumed in
the derivation given in the Appendix for several reasons. First, the
square law device serves to reduce low-level interference in the absence
of pulses. In this way, the adverse pulse density effects are reduced. Sec-
ondly, the square law assumption is a good approximation to a combina-
tion of slicer and rectifier that is actually used in the physical embodi-
ment of this scheme. Finally, the square law characteristic is somewhat
more convenient for analysis than the actual symmetrically biased recti-
fier.
The reader familiar with the literature will recognize that in the case
p = 1/2, the spectrum of bipolar is identical with Meacham's twinned
binary. 14 This follows from the fact that in Meacham's approach, twinned
binary is generated by taking the unipolar train, delaying it by one time
slot, and subtracting it from the original. This modifies the spectrum of
unipolar by | 1 — <C JuT | a = 2(1 — cos co7"), which is identical to the
modifying factor of (13) for p = 1/2. Despite the fact that the spectra
arc identical for equally likely unipolar pulses and spaces, the relation-
ships between the original unipolar code and the transmitted code differ
for the two methods. If the probability of error (both insertion and dele-
fion) for the unipolar train is P c for interference that is equally likely to
be positive and negative, then it can be shown that the error probabilities
for bipolar and twinned binary are (3/2)P e and 2P e respectively. The
above figures assumed that the repeaters to reconstruct either train do
not have the bipolar constraint built in. In addition, due to the manner
in which the twinned binary is converted to binary at the receiver, dou-
ble errors per word are much more likely than in bipolar. For purposes
of classification, bipolar belongs to the class of codes in which the con-
version from unipolar is digital and the reconversion is analogue. In
twinned binary the reverse is true.
There are two points of departure from the bipolar converter of Fig.
* A timing component could be added at the spectral null at the bit rate. This
would not help the crosstalk timing problem except for the adverse pulse density
effect .
t Obviously energy for timing does not come from a single frequency. However
the characterization is convenient. Further clarification of this point is given in
the Appendix and Section VI.
128
THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1902
11. Increasing the number of counter stages to N results in a code in
which N successive unipolar pulses are transmitted positive and the
next N, wherever they occur, are transmitted negative. Spaces are unaf-
fected. We have called such a pseudo-ternary code JV-pulse. For N = 2,
p = 1/2, and balanced pulses, the power spectrum has a continouus
component only, given by:
p( f \ _ 4 (! - coscoT)(3 - 2coswT)
Pi(/).
(14)
5-12 cos a>T + 8 cos 2 uT
The spectrum is shown in Fig. 12 for 50 per cent duty cycle rectangular
pulses. The power spectrum under similar conditions for N = 3 is also
shown in this figure. As before, rectification can be used to convert back
to unipolar.
In JV-pulse, for dense pulse patterns the low-frequency suppression
problem is more severe than in straight bipolar. Furthermore, the spec-
trum peaks up at low frequencies, thereby increasing the interference
sprayed into AM systems using pairs in the same cable. Therefore, for
this application, JV-pulse is inferior to straight bipolar.
5.4 Higher-Order Bipolar
Another possible direction, first suggested by M. Karnaugh, involves
paralleling bipolar converters and routing the unipolar pulses to each of
Fig. 12 — N pulse power spectra.
PCM TRANSMISSION IN THE EXCHANGE PLANT
129
the converters in sequence. Such a composite converter is shown in Fig.
13. We will only give results for the power spectrum for balanced pulses
and p = 1/2. With these conditions, for N converters in parallel the
power spectral density is
/>(/) = 2[1 - cos Na>T]P 1 (f) l
(15)
Rectification suffices to reconstruct the original binary train.
The most interesting realization for our purposes is interleaved bipolar
i.e., N = 2.It can be seen from (15) that the spectrum contains nulls at
dc and integer multiples of half the bit rate. This permits the addition of
a sinusoidal component at half the bit frequency to the pulse train for
purposes of timing. In this manner, timing jitter due to finite pulse width
effects is ideally removed, as is the adverse pulse density penalty. These
are the principal features of this approach. As in 2-pulse, two pulses in
a row may be of the same sign, thereby roughly doubling the low-fre-
quency tail in the succeeding time slot and reducing tolerance to cross-
talk. The addition and extraction of the sinusoidal component from the
pulse train is not an easy nor an inexpensive circuit problem, and in
practice results in an inevitable interaction between the pulse train and
the sinusoidal signal to impair the decision-making process in the re-
peater. Specifically, balance requirements on the positive and negative
pulses are particularly severe to suppress the discrete component in the
pulse spectrum. The added timing component can be made sufficiently
large to overcome the unbalances. However this increases the power-
handling requirements of the repeater. Complete retiming and pulse
width control are more difficult to implement with this approach than
in bipolar for 50 per cent duty cycle pulses. The continuous spectrum at
UNIPOLAR PULSES IN
CLOCK OR DIGIT COUNTER
TO CONTROL SWITCH
(TRANSFER GATE)
*BIP- BIPOLAR CONVERTER
OF FIGURE II
Pig. 13 — Bipolar N block diagram.
130 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1962
f r /i) is [1 - cos (4ir/6)]/[l - cos (2x/6)] = 3 times that of bipolar,
thereby making compatability between N carrier and PCM more diffi-
cult. For 100 per cent duty cycle pulses, complete retiming and width
control are even more difficult, and the interference into N carrier is
increased by another factor of 4* over 50 per cent duty cycle bipolar.
All of the above factors point to bipolar as the preferable scheme pro-
vided that the timing jitter accumulation is satisfactory for the system
lengths under consideration. This turns out to be the case. 3
From the form of (15) it can be seen that Meacham's twinned binary
can be extended to realize the same spectrum simply by delaying the
unipolar train N time slots prior to subtraction. The reconversion at the
receiver consists of N parallel converters, each similar to the twinned
binary converter. This is an extension of the previous classification; i.e.,
bipolar-A/ involves N parallel digital converters at the transmitting
terminal and a single analogue restorer at the receiver, while the exten-
sion of Meacham's twinned binary consists of an analogue operation at
the transmitting terminal and N parallel digital restorers at the receiving
terminal.
5.5 Reduction of Adverse Pulse Density Penalty
Before we go on to summarize the various pseudo-ternary codes given
above, and their realization, we should pause to consider methods for
reducing the adverse pulse density penalty. Several means are available,
some of which have already been indicated. They include:
1 . Use of a slicer to prevent interference from entering the tuned cir-
cuit in the absence of a desired pulse.
2. Addition of a timing wave at a spectral null.
3. Use of a very high Q timing circuit to integrate over a long time
interval.
4. Addition of noise or a tone in an idle channel.
5. Homogenizing the code by using alternate interchange. 16
The first three techniques given above relate to repeater design, while
the last two methods involve terminal processing to minimize the prob-
lem at the source. Addition of noise or a tone to an idle channel or al-
ternate interchange are believed to be too expensive for this application.
Furthermore, these approaches will not be useful for data transmission.
Item 3 above is included to point out that the adverse pulse patterns are
transient effects that can be smoothed out by narrow timing filters. Too
high a Q is precluded by previous considerations of stability of the tuned
* This assumes that the transmitted pulse peaks arc identical in both cases.
PCM TRANSMISSION IN THE EXCHANGE PLANT 131
circuit. The remaining items have been considered under bipolar and
interleaved bipolar respectively.
5.6 Other Timing Approaches
Use of the dual-mode scheme (listed in Table I), with timing informa-
tion transmitted in only one direction, can in principle avoid the cross-
talk timing problem. However, delay differences between pairs requires
delay build-out of the various links, which is akin to the magnitude of
the pair selection problem. Similarly, we have devised separate pair
timing schemes in which the NEXT problem, as far as timing is con-
cerned, is ideally eliminated, and of course pattern jitter is removed.
Again this implies the provision of several clock phases to account for
differences in delay among the pairs in the cable. Variations in frequency
and phase among the terminals and the transmitted clock will reduce
tolerance to pulse crosstalk that obscures the decision-making process.
In addition, an economic penalty results from utilizing a separate pair
for timing. While it is possible to reduce the economic penalty by sharing
a single clock among several pairs, this introduces a reliability problem
that again is translated into increased cost. For these reasons a self-timed
approach is more suitable.
5.7 Choice
Other self-timing approaches have been considered for this applica-
tion. Some are variants of the schemes discussed, and all are inferior in
most respects to those previously enumerated.
From the standpoint of timing alone, "idealized" interleaved bipolar
with timing wave added at/ r /2 is the best approach. When we consider
the problems of realization, economics, compatibility with N carrier,
and crosstalk into the information-bearing path this approach falls down
the preference ladder below bipolar.
Bipolar is chosen for this application since it has the following advan-
lafjes:
1. Energy for timing comes from a region of lower attenuation and
higher crosstalk loss than in unipolar.
2. Low-frequency suppression problems are minimized, thereby re-
laxing transformer requirements and cost.
3. Slicing in the timing path reaps two rewards, namely (a) low-level
interference in the absence of a pulse is removed; this reduces the adverse
pulse density penalty; and (b) pulses exciting the tuned circuit are nar-
rower; consequently finite pulse width effects are reduced.
132 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1902
4. Turnover problems associated with connecting the unbalanced re-
peater to the balanced line are eliminated.
5. A simple error meter can and has been devised to count bipolar
violations and monitor the state of health of the line.
6. Relative compatibility with N carrier is as good or better than in
other schemes considered.
Most important, however, is the fact that all of these features plus
complete retiming and pulse width control can be translated into a physi-
cal repeater with presently available components.
VI. PULSE SHAPING (EQUALIZATION)
Most of the foregoing material has dwelt on the problem of timing in
self-timed repeaters subject to severe crosstalk interference. Even in the
absence of timing interference, it should be clear that crosstalk interfer-
ence will reduce the ability of a repeater to make a pulse-no pulse deci-
sion. Furthermore, the effect of crosstalk in causing errors will be de-
pendent upon the equalization employed in the repeater. In reality, the
effects of crosstalk on both timing and pulse recognition cannot be di-
vorced. Ideally, we would like to be able to synthesize the complete
repeater structure from input-output specifications. Unfortunately, the
present state of the art is such that this is not possible. For this reason
we address ourselves to a simpler question. What pulse shaping (both
linear and nonlinear) should be employed such that a perfectly timed
threshold detector can operate on the incoming signal to reconstruct the
desired signal with a minimum probability of error? Since the statistics
of the interference are not sufficiently well known, we cannot answer this
question. Therefore, we lower our sights further and specialize to the
case where the over-all equalized medium (cable plus equalization) is
linear and the desired pulse presented to the threshold circuit belongs to
a specified family. Several pulse shapes are suitable: i.e., linear phase
low-pass Gaussian (LPG), raised cosine, or time response of maximally
flat delay transfer function. We will use the LPG simply because the
analysis is more tractable and because it can be approached with realiza-
ble circuitry. If we also use the lumped capacitor and single interferer to
represent the crosstalk interference, we can (subject to certain simplifi-
cations) determine that member of the LPG family that minimizes the
error probability for a specified coupling. Alternatively, we can consider
the "worst-case eye" assumed by Mayo. 3 Under this condition, we can
find the parameter of the LPG characteristic such that the sum of inter-
symbol interference plus peak crosstalk is a minimum for a preselected
PCM TRANSMISSION IN THE EXCHANGE PLANT 133
crosstalk loss. An LPG characteristic which is 6 db down at half the
pulse repetition frequency gives close to the optimum performance with
an allowable crosstalk loss of about 30 db that just closes the eye. De-
tails of this exercise in the application of the Fourier transform are not
covered herein. The analysis is straightforward but messy. With the
addition of low-frequency suppression, mistuning, finite width of the
sampling pulse and threshold variations to intersymbol interference, it
is not surprising that the allowable equivalent crosstalk loss must be
raised to about 35 db to just cause errors in the real life repeater. This
allowable coupling must be increased for lines longer than nominal and
other factors discussed in Mayo's paper. In passing we note that the
actual equalized pulse at the output, of the repeater under nominal con-
ditions agrees closely with the response of the optimum LPG characteris-
tic to a 50 per cent duty cycle rectangular pulse.
The spectrum of the bipolar pulse train peaks up at odd integer mul-
tiples of f r /2. Peaking in the neighborhood of (3/2)/ r can be troublesome
because energy in this region transmitted via the crosstalk path can
beat with the desired timing component in the rectifier to produce addi-
tional interference at the timing frequency. Therefore, it is desirable to
reduce the preamplifier gain in this region. This feature is included in
the repeater. 3 Attendant to this modification is a slight change in inter-
symbol interference. The most pronounced advantage of this feature
occurs for dense interfering pulse trains.
In principle, equalization in the timing path can profitably differ
from that employed in the information path in the repeater. By the
analysis outlined in the Appendix (Section A.2), it can be concluded
that the pulses presented to the rectifier should be essentially the same
as those giving optimum performance in the information path if the
criterion is that the desired bit rate component at the rectifier output
be twice that of the undesired component. This result is based on a ran-
dom crosstalk interferer through a capacitive path and a desired pulse
train that is random. In addition, this analysis permits us to conclude
that under random conditions, crosstalk into the timing path should
not be limiting, but crosstalk effects on the information bearing path
are limiting. This improvement over the half bit rate figure used in
Section V is due to the different spectral content of the desired pulse
train and the crosstalking train that enter the square law rectifier
assumed in the analysis. In this regard the bipolar repeater is superior
to interleaved bipolar with timing wave added. Furthermore, it should
be realized that this .advantage is even greater for a dense desired pulse
train and ;i sparse interferer. < )n the other hand, lor adverse pulse
134 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1962
densities in periodic patterns the converse is true — timing becomes
limiting before the pulses are obscured by crosstalk. These conclusions
agree with experiment.
Before we bring this paper to its conclusion, it might be worthwhile
to make a few remarks about our philosophy for choosing a transmission
scheme for this application. In many respects we have focused our atten-
tion on worst cases. Attendant to this approach is the inherent danger
of over-engineering. That we are free from this accusation follows from
the fact that the resulting repeater satisfies economic objectives, size
requirements, and permits considerable growth of the exchange plant.
Furthermore, even if we neglect timing interference, other factors would
dictate a repeater of virtually the same complexity. Indeed a carefully
designed unipolar repeater is not significantly simpler than the bipolar
repeater and suffers more loss in margin due to low-frequency sup-
pression.
There has been a tendency in the literature on the design of PCM
repeaters to consider only the mean square value of each source of inter-
ference. This is generally followed by the addition of the mean square
values of all of the sources of interference and a statement that the re-
sulting distribution of the sum is normal with mean square value given
by the sum of the respective mean square values. In effect, the central
limit theorem is invoked by incantation, not by proof. Furthermore even
if the resulting distribution is normal in the neighborhood of the mean,
this says nothing about the behavior in the neighborhood of the tails.
This is the region of interest in most high-quality pulse systems. In point
of fact it can be shown that most of the sources of interference con-
sidered herein have distributions that deviate considerably from the
normal and are skewed adversely. We will demonstrate this for timing
jitter due to mistiming and finite pulse width in another paper. 17 There-
fore the treatment of the sum of the sources of interference as being nor-
mal involves a dangerous pitfall that we have studiously avoided.
VII. DISCUSSION
With the exception of some model building in the Appendix (Section
A.2) we have dealt with essentially deterministic extrapolations rather
than with true stochastic models of the real world. As noted previously,
this simplification was essential in order to proceed with the develop-
ment. Subject to these and other assumptions we have shown that the
bipolar transmission scheme is well suited to the exchange plant environ-
ment. With a random crosstalker and a random desired signal, bipolar
has a crosstalk figure of merit for worst case line loss about the same as
PCM TRANSMISSION IN THE EXCHANGE PLANT 135
TPC, about 35 db. This figure is about 23 db better than that for uni-
polar under the same conditions. Tn addition, slicing in the timing path
goes a long way toward eliminating the adverse pulse density penalty
of 17 db.
The crosstalk figure of merit of 35 db is well below the 1 per cent loss
point on the distribution given in Fig. 7. Despite the preliminary nature
of these data, this crosstalk figure indicates that it certainly is possible
to place a single PCM system within a unit of a unit-constructed cable
with no pair selection. Indeed it should be possible to accommodate
several systems. Neither of these statements can be made for unipolar.
However, the question of how many systems can be installed cannot be
answered quantitatively from the work reported in this paper. The
multi-system performance is dependent upon how the crosstalking inter-
ferers add. This difficult statistical problem is presently being considered
both analytically and experimentally by the Systems Engineering De-
partment at Bell Telephone Laboratories. Fortunately, there are a host
of techniques that can be employed to further improve the crosstalk
situation. They all imply an economic penalty and involve special sys-
tem layouts. For example, shorter repeater spacing and staggering re-
peater locations for different directions of transmission can appreciably
reduce crosstalk interference. Of course, where it is possible to use
separate units or separate cables for the two directions of transmission,
this will significantly decrease the probability of crosstalk-induced
failure.
We can add nothing to what we have already said about compata-
bility with X carrier. This question is still under study. Here again
special measures can be adopted to combat the problem.
With regard to impulse noise, extensive measurements made by T. V.
Crater indicate that this should not be limiting provided that the re-
peater spacing from an office is about half of that of the normal line
repeaters.
In short, further consideration of the interaction of the plant with the
bipolar repeater is receiving considerable attention by the Systems
Engineering Department at Bell Laboratories. Therefore a more quan-
titative picture will have to await the outcome of their work.
vin. Conclusion
Based largely on timing considerations we have shown that the con-
ventional unipolar PCM transmission scheme is unsuitcd to the Ex-
change Plant where crosstalk interference is the principal transmission
deterrent. Several pulse transmission schemes have been examined for
136 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1902
replacement of unipolar. We have chosen a bipolar transmission scheme
and a self-timed reconstructive repeater incorporating nonlinear timing
wave extraction with complete retiming and pulse width control. Rea-
sons for making this choice have been covered in Section V. For random
pulse trains on the disturbed and disturbing pairs, bipolar virtually
eliminates crosstalk timing interference as a contributor to failure of a
repeater.
IX. ACKNOWLEDGMENTS
Many of the concepts and ideas put forth in the paper arose out of
useful discussions with J. S. Mayo and E. E. Sumner. It is a pleasure
to acknowledge their contributions to this effort.
APPENDIX
Bipolar Through a Square Law Device
a.i Random Signal Alone
The signal transmitted in bipolar can be represented by
y(t) = E a„g(t-nT) (16)
where a n is a random variable taking on the values - 1, 0, 1 with a priori
probabilities p/2, 1 - p, and p/2 respectively. It is readily shown that
ave y(t) is zero. When this pulse train is put through a square law
device, we have at its output
y\t) = \ E Wit - nT)T
= 2 a^fit - nT) + £ S a n a m g(l - nT)g(t - mT).
n^i
The first term above is due to the squares of individual pulses, while the
second term arises due to pulse overlaps. The average value of the
squared pulse train is
ave y\t) = £ R(0)g\t - nT)
+ IE R(k)g{t - nT)g(t - (n + k)T) (1»)
with
R(k) = ave a„a n+k . (19)
PCM TRANSMISSION IN THE EXCHANGE PLANT 137
That the (average) output of the square law device is periodic may be
shown by replacing t by t + T in ( 18) to get
ave y\l + T) = Z R(0)g 2 (t - (n - 1)T)
n
+ Z Z /«(%(« - (n - Dr)^(* - (n- l + Jb)D , om
= ave i/ 2 (0
with appropriate changes in summation indices.
Since the ensemble average is periodic, it can be expanded in a Fourier
series. Following the same procedure used by Bennett 2 we get
00
avey 2 (<) - Z C m exp (j2rmrf r t) (21)
n=— oo
with
C m = /, ffl(O) /°° /(i*)a-*-^-«lu
'—CO
r i (22)
+ Z B(*0 / r/(w)f/(w + kT)e- flm/rU du
k •'-oo
The prime on the summation over k indicates that the k = term is to
be omitted.
Our main interest resides in the component at the bit frequency,
namely C\ or C-\ . By using (he relationship between the Fourier trans
form of the product of two functions and the convolution of their trans-
forms, we obtain
C, = /, \r(0) f G( -/)<?(/, + /) df
r i (23)
+ Z «(*) ) m G( -/)<?(/, + f)e* wk,T df\ .
Since R (k) = R(-k), (23) becomes
Cx = f r jf G( -f)G(f r + /) [r(0) + 2 Z fl(Jfe) cos 2 7 r/,-/2 , l # (24)
Multiplying numerator and denominator of the integral by 0(f) gives
c '-£t^T p -w* (25>
where we have taken advantage of the fact that Pa(/) = P 3 (—f) and
138 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1902
P 3 (/) = /, I 0(f) I 2 1/2(0) + 2 g B(fe) cos 2ttA-/t| . (26)
It will be recalled from the text that P 3 (f) is the label attached to the
bipolar spectrum. That the power spectrum of bipolar is given by (26)
follows from the general relationship derived for the spectrum of a
digital source given by Bennett 7 or from Wolds' theorem for the power
spectrum of a stationary time series. 16 Alternatively it can be shown that
R(k) = (-l)'V(2p - l) |tM |fc| > 1
R(l) = -p 2 (27)
R(0) = p.
Substitution of these expressions for R(k) in (26) yields the expression
for P 8 (/) given in (12).
To give a graphic picture of the frequency region that contributes
most to the bit frequency component, we make the following assump-
tions:
1. The pulse shape is Gaussian with transform given by the low-pass
Gaussian characteristic below
G(f) = exp [-0.693 hQ j
where / 6 is defined as the frequency at which the response is 6 db down
from its low-frequency asymptote.
2. p = 1/2; therefore R(k) = for \k\ > 1. Under these assump-
tions (25) becomes
Ci = \ \\ exp h±p [(/ - frf + /]} (1 - cos 2tt/T) df. (29)
The integrand of (29) (neglecting a multiplicative constant) is plotted
in Fig. 14 for / 6 = / r /2. From this figure it is apparent that the major
contribution to the integral comes from the neighborhood of f r /2.
a. 2 Random Signal Phis Random Crosslalkers
If we consider the addition of several crosstalkers to the desired signal,
the composite signal exciting the square law characteristic is
z(t) = nil) + x(t) (30)
where ij(t) is given by ( 16) and the crosstalk interference x(t) is given by
(28)
PCM TRANSMISSION IN THE EXCHANGE PLANT
139
0.4
0.3
0.2
0.1
-0.6
1
(\
i
\
\
1
\
J
V
-0.4
0.6
£
fr
1.0
1.6
Fig. 14 — Integrand of (25) (except for a constant).
•(t) = E I b h Mt - nTi - Ti )
(31)
for N interferers. It is assumed that the b in are independent random vari-
ables with identical distributions, each distributed in the same manner
as the a„ of the desired message. The function hi(t) is the response of the
ith crosstalk path and the preamplifier of the disturbed repeater to a
PCM pulse; r, is a measure of the phase difference between the ith
crosstalk pulse train and the desired message; and Ti is the reciprocal of
the pulse repetition frequency of the t'th pulse train. If we let 7\ =
T + A, , then A, is a measure of the difference in pulse repetition fre-
quency between the ith crosstalker and the desired message. In general
hi(t), n , and A, are random variables with the latter two being slowly
time varying. We will assume the same frequency for all interferers;
therefore A, = for all i. When we average z(t) over the a and b en-
sembles, we find that its average value is zero as expected. At the output
of the square law device, the average value of z 2 (t) is
iivez'it) = ave ?/-(/) + ave x\t)
(32)
since x(t) and y(t) are independent random variables with zero mean
(with respect to averaging over the message ensembles). By the same
argument used previously, ave z*(t) is periodic with period T and can
140 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1062
be expanded in a Fourier series. The component at the pulse repetition
frequency, di , is
d t = C x + Ci. (33)
where
C = t e j2 "' Ti f H i f ( " ( r) PM) df (34)
, = i J- x Hi{—J)
and Ci is given by (24). P 3 »(/) is the power spectrum of the ith cross-
talker at the input to the square law device, and is given by P 3 (f) when
Hi(f) is substituted for G{f).
Let us assume that each Hi = A t H. In this context, H is the transform
of a "representative" crosstalk pulse and the 4, are random variables.
With this relationship substituted in (33), we get
AT
z
(=1
(35)
ff(/ P - f)H(f)[R(0) + 2 X «(*) cos 2x*/T] <*/.
■/:
From (33) we can write the average timing wave component as
w(t) = 2 | d | cos (2tcM + »i ) + 2 | Cu I cos (2*/ r « + 0*) (3G)
where 0i and 0* are the angles of C\ and C u respectively. The representa-
tion of (34) enables us to define an "equivalent crosstalk interferer,"
namely
x(t) e =
Za
2 JUfrTi
1/2
£ MO - nT - t). (37)
In (37) r is a random variable chosen to have the same distribution as
the random component of 0*/27r/ r . Of major interest is the magnitude
of the equivalent interferer. If we assume that H(f) represents the fre-
quency dependence of the "smooth" crosstalker as modified by the char-
acteristic of the equalizer of the disturbed repeater, then we can assume
that the A t are random variables with the distribution of Fig. 7 that
serve to change the level of the smooth characteristic. This model is
useful primarily to give some idea as to how the crosstalkers combine to
interfere with the desired timing component. We are still left with the
difficult problem of determining the distribution of a complicated func-
tion of approximately log-normally distributed random variables in
order to determine how the ensemble of average timing waves varies
PCM TRANSMISSION IN THE EXCHANGE PLANT 141
with crosstalk interference. This of course does not indicate how the
crosstalk interferers add to confuse the threshold circuit in the repeater.
Again we must lower our sights and revert back to the equivalent
capacitive coupling in order to get some feel for how equalization affects
the timing path. For an over-all LPG characteristic we can compute the
allowable crosstalk coupling such that the ratio of desired to undesired
timing component is 2 corresponding to a maximum of 30° phase shift
in the tuned circuit; i.e. | C\ |/| C lz \ = 2. The maximum allowable cross-
talk will be a function of the equalization as characterized by the 6-db
point on the LPG characteristic. Evaluation of the integrals for Ci and
Ci x (as modified for the deterministic capacitive coupling) shows that
the optimum is rather broad and occurs in the neighborhood of / 6 = / r /2.
Furthermore the allowable crosstalk coupling is a few db larger than the
allowable crosstalk coupling to close the eye in the absence of timing
interference. Therefore under random pulse conditions, timing is not
limiting.
Similar computations can be made for periodic pulse patterns. This
has been done to check the experimental results given by Mayo 3 in his
Fig. 57. Analytical results are within 2 db of measured results for the
patterns that were spot checked.
IlEFEKENCES
1. Davis, C. G., this issue, p. 1.
2. Mann, H., Strauhe, H. M., and Villars, C. P., (his issue, p. 173.
3. Mayo, J. S., this issue, p. 25.
4. DeLange, O. E., B.S.T.J., 36, Jan., 1956, p. 67.
5. Wrathall, L. R., B.S.T.J., 35, Sept., 1956, p. 1059.
6. Sunde, E. D., B.S.T.J., 36, July, 1957, p. 891.
7. liennetl, W. R.. H.S.T.J.. 37, Nov., 195S, p. 1501.
8. Richman, D., Proc. I.R.E., 42, Jan., 1954, p. 106.
9. De Lunge, O. E., and Pustelnvk, M., B.S.T.J., 37, Nov., 1958. p. 1487
10. Rowe, H. E., B.S.T.J., 37, Nov., 1958, p. 1543.
11 . Bennett, W. R., Proc. Symposium on Modern Network Synthesis, 5, 1956, p. 45
12. Huggins, W. H., Proc. I.R.E., 45, Jan., 1957, p. 74.
13. Zadeh, L. A., Proc. I.R.B.. 45, Oct., 1957, p. 1413.
14. Meacham, L. A., Twinned Binary Transmission, U.S. Patent No. 2,759,047.
15. Carhrey, R. L., Proc. I.R.E., 48, Sept., 1960, p. 1546.
16. Wold, iVI., .4 Study in the Analysis of Stationary Time Series, Dissertation
Upsala, 1938.
17. Aaron. M. R., and Gray, J. R., to be published.
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